where \(\mathrm {log}L(\hat{\theta })\)  is the log-likelihood
Equation REF  evaluated to model-implied covariance matrix,
\(\Sigma (\hat{\theta })\)  and \(\mathrm {log}L(\theta _\mathrm {max})\)  is the
log-likelihood for an exact fit; i.e., \(\Sigma (\hat{\theta }) = S\) .
P-values are derived either from the \(\chi ^2(\mathrm {df})\)  distribution
with \(\mathrm {df} = p(p + 1)/2 - t\)  degrees of freedom, or from a resampling
bootstrap distribution [1]}. Non-significant P-values
(P > 0.05) indicate that the model provides a good fit to data (i.e.,
the elements of \(S - \Sigma (\hat{\theta })\) , should be close to zero).
Alternatively to the chi-square test, the Akaike's
information criterion (AIC) [2]} can be used to compare fitted
to saturated model, defined in SEM as [3]}:
\( \mathrm {AIC} \, = \, -2 \, \mathrm {log} L(\hat{\theta }) + 2t \, \approx \,\chi ^2 - 2 \mathrm {df}\) 
